I don't know Terry Pratchett's thinking behind the character, but Carrot strikes me as a reductio ad absurdum of the concept.

I don't think it's a reductio. Actually, I think it's almost the opposite; one of Pratchett's usual schticks is drawing up exaggerated social and political concepts that look absurd on their faces but later turn out to make internal sense. Looked at in that light, it's pretty clear what's going on: Carrot isn't phenotypically a dwarf, but he's culturally a dwarf, and he's accepted as such by Pratchett's dwarves because, to them, dwarvishness is less about being short and beardy and more about the culture. Wearing mail to dinner, being intimately familiar with mine engineering, baking bread that doubles as an assault weapon, et cetera. I don't know for sure, but I suspect the situation Pratchett had in mind was something like being adopted into a religious group normally associated with another ethnicity: his dwarves very often play on religious and traditional themes.

As Pratchett's developed Discworld's dwarvish culture more, this has started to make less sense, but the series isn't particularly good at long-term thematic continuity.